waiting line model

1. A
PROJECT BASED LAB REPORT
On
SOLVING OF WAITING LINES MODELSINTHERESTAURENT USING QUEUEING
THEORY MODEL
In the course of operation research
Submitted in partial fulfilment of the
Requirements for the award of the Degree of
Bachelor of Technology
In
MechanicalEngineering
By
Subham kumar gupta(14007582)
DEPARTMENT OF MECHANICAL ENGINEERING
(DST-FIST Sponsored Department)
K L University
Green Fields, Vaddeswaram, Guntur District-522 502
2016-2017
K L University
DEPARTMENT OF MECHANICAL ENGINEERING

2. This is to certify that this project based lab report entitled “SOLVING OF WAITING LINES MODELS
IN THE RESTAURENT USING QUEUEING THEORY MODEL ” is a bonafide work done by subham
kumar gupta (14007582) in partial fulfillment of the requirement for the award of degree in
BACHELOR OF TECHNOLOGY in Mechanical engineering during the Acadamic year
2016.
Lecturer In Charge Head of the Department
Dr.sanjaykrishna Dr.S S RAO
CERTIFICATE

3. K L University
DEPARTMENT OF MECHANICAL ENGINEERING
We hereby declare that this project based lab report entitled “SOLVING OF WAITING LINES
MODELS INTHE RESTAURENT USING QUEUEING THEORYMODEL ” has been prepared
by us in partial fulfilment of the requirement for the award of degree “BACHELOR OF
TECHNOLOGYin MECHANICAL ENGINEERING” during the academic year 2016-2017.
I also declare that this project based lab report is of our own effort and it has not been submitted
to any other university for the award of any degree.
Date: Name
Place:Vadeswaram SUBHAM KUMAR GUPTA(14007582),
DECLARATION

4. Abstract:
Waiting lines and service systems are important parts of the business world. In this project we
describe several common queuing situations and present mathematical models for analyzing
waiting lines following certain assumptions. Those assumptions are that (1) arrivals come from
an infinite or very large population, (2) arrivals are Poisson distributed, (3) arrivals are treated on
a FIFO basis and do not balk or renege, (4) service times follow the negative exponential
distribution or are constant, and (5) the average service rate is faster than the average arrival
rate. The model illustrated in this Restuarent for customers on a levelwith service is the multiple-
channel queuing model with Poisson Arrival and Exponential Service Times (M/M/S). After a
series of operating characteristics are computed, total expected costs are studied, total costs is
the sum of the cost of providing service plus the cost of waiting time. Finally we find the total
minimum expected cost.

5. ACKNOWLEDGEMENTS
My sincere thanks MR.SANJAYKRISHNA Sir in the Lab for their outstanding support
throughout the project for the successful completion of the work
We express our gratitude to Dr.S S RAO, Head of the Department for Mechanical Engineering
for providing us with adequate facilities, ways and means by which we are able to complete this
project work.
We would like to place on record the deep sense of gratitude to the honourable Vice Chancellor,
K L University for providing the necessary facilities to carry the concluded project work.
Last but not the least, we thank all Teaching and Non-Teaching Staff of our department and
especially my friends for their support in the completion of our project work.
Name
Subham kumar gupta(14007582)

6. INTRODUCTION
OperationresearchisscientificknowledgethroughInter-disciplinaryteamworkforpurpose of
determiningthe bestutilizationof limitedresources.Itisanapplicationof scientificmethods
techniquesandtoolstoproblemsinvolvedin operationof systemsastoprovide these incontrol
of operationswithoptimumsolutionstoproblem.ORisan analytical methodof problemsolving
and decisionmaking. Setof acts requiredforthe achievementof adesiredoutcome iscalledas
an operation.
Operational research(OR) encompassesawide range of problem-solvingtechniquesand
methodsappliedinthe pursuitof improveddecision-makingandefficiency,suchassimulation,
mathematical optimization,queuingtheoryandotherstochastic-process models,Markov
decisionprocesses,econometricmethods,dataenvelopmentanalysis,neuralnetworks,expert
systems,decisionanalysis,andthe analytichierarchyprocess.Nearlyall of these techniques
involve the constructionof mathematical modelsthatattempttodescribe the system.Because
of the computational andstatistical nature of mostof these fields,ORalsohasstrongtiesto
computerscience andanalytics.Operational researchersfacedwithanew problemmust
determine whichof these techniques are mostappropriate giventhe nature of the system, the
goalsfor improvement,andconstraintsontime andcomputingpower.
Limitationsof OR:
1. Mathematical models are applicable to only specific category of problems.
2. Being a new field, generally there is a resistance from employees to new proposals.
3. Management who has to implementthe advised proposals may itself offer a lot of resistance
due to conventional thinking.
4. Young & enthusiastic overtaken by its advantages & exactness
Advantages of OR:

7. 1. Better co-ordination
2. Better planning.
3. Flexibility
4. Better decision making
5. Improves productivity
Disadvantages of OR:
1. Requires trained people
2. Lengthy Procedure
3. Complex in solving problems
4. Time taking
5. Initial investment is high for fulfilling needs & requirements.
Modelsin OR:
It isdefinedasanideal representationof real lifesituationwhichrepresentsone (or) afew
aspectsof reality
Modeling
By Method By Purpose By Structure By Nature By Behaviour

8. Applicationof OR:
Applicationsof managementscience isabundantinindustryasairlines,manufacturing
companies,serviceorganizations,militarybranches,andingovernment.The range of problems
and issuestowhich managementscience hascontributedinsightsandsolutionsisvast.It
includes:
• scheduling airlines, including both planes and crew,
• deciding the appropriate place to site new facilities such as a warehouse, factory or fire
station,
• managing the flow of water from reservoirs,
• identifying possible future development paths for parts of the telecommunications
industry,
• establishing the information needs and appropriate systems to supply them within the
health service, and
• identifyingandunderstandingthe strategiesadoptedbycompaniesfortheirinformation
systems
Some otherrelatedfieldsinwhichoptimisationcanbe done usingOperationsresearch
techniquesare:
1.Agriculture 9.BusinessAnalytics 17. Logistics
2.Finance 10.Data mining 18. Supply chain
management
3.Marketing 11.Decisionanalysis 19.Stochasticprocesses
4.Personal management 12. Engineering 20. Simulation
5.Production
management
13.Financial engineering 21. Policyanalysis
6.Purchasing 14. Game theory 22.Projectmanagement

9. 7.Facilitiesplanning 15. Graph theory 23.Mathematical
optimization
8.R & D 16.Industrial engineering 24. Social network
Techniquesusedin Operationresearch:
1. Assignment problem
2. Decision analysis
3. Dynamic programming
4. Inventory theory
5. Linear programming
6. Mathematical optimization
7. Optimal maintenance
8. Queuing theory
9. Real options analysis
10. Stochastic processes
11. Systems analysis
12. Systems thinking
13. Transportation
14. Game theory
15. Simulation.
16. Project management 17. PERT/CPMtechnique
18. Scheduling.
Introduction ofQUEEING THEORY:-
A flow of customers from finite or infinite population towards the service facility forms a queue
(waiting line) an account of lack of capability to serve them all at a time. In the absence of a
perfect balance between the service facilities and the customers, waiting time is required either
for the service facilities or for the customers arrival. In general, the queueing system consists of
one or more queues and one or more servers and operates under a setof procedures. Depending
upon the server status, the incoming customer either waits at the queue or gets the turn to be

10. served. If the server is free at the time of arrival of a customer, the customer can directly enter
into the counter for getting service and then leave the system. In this process, over a period of
time, the system may experience “ Customer waiting” and /or “Server idle time”
1.1 Queueing System:
A queueing system can be completely described by
(1) the input (arrival pattern) (2) the service mechanism (service pattern) (3) The queue
discipline and (4) Customer’s behaviour
1.2. The input (arrival pattern)
The input described the way in which the customers arrive and join the system. Generally,
customers arrive in a more or less
random manner which is not possible for prediction. Thus the arrival pattern can be described
in terms of probabilities and
consequently the probability distribution for inter-arrival times (the time between two
successive arrivals) must be defined.
We deal with those Queueing system in which the customers arrive in poisson process. The
mean arrival rate is denoted by ƛ.
1.3 The Service Mechanism:-
This means the arrangement of service facility to serve customers. If there is infinite number of
servers, then all the customers are served instantaneously or arrival and there will be no queue.
If the number of servers is finite then the customers are served according to a specific order
with service time a constant or a random variable. Distribution of service
time follows ‘Exponential distribution’ defined by f(t) = µe -µt , t > 0 The mean Service rate is
E(t) = 1/µ
1. Queueing Discipline:-
It is a rule according to which the customers are selected for service when a queue has been
formed. The most common disciplines are
1. First come first served – (FCFS) 2. First in first out – (FIFO) 3. Last in first out – (LIFO) 4.
Selection for service in random order (SIRO)
1. Customer’s behaviour
1. Generally, it is assumed that the customers arrive into the systemone by one. But in some
cases, customers may arrive in groups. Such arrival is called Bulk arrival.

11. 2. If there is more than one queue, the customers from one queue may be tempted to join
another queue because of its smaller size. This behaviour of customers is known as jockeying.
3. If the queue length appears very large to a customer, he/she may not join the queue. This
property is known as Balking of customers.
4. Sometimes, a customer who is already in a queue will leave the queue in anticipation of
longer waiting line. This kind of departare is known as reneging.
1.6 List of Variables
The list of variable used in queueing models is give below:
n - No of customers in the system C - No of servers in the system
Pn (t) – Probability of having n customers in the systemat time t.
Pn - Steady state probability of having customers in the system
P0 - Probability of having zero customer in the system
Lq - Average number of customers waiting in the queue.
Ls - Average number of customers waiting in the system (in the queue and in the service
counters)
Wq - Average waiting time of customers in the queue.
Ws - Average waiting time of customers in the system (in the queue and in the service counters)
ƛ- Arrival rate of customers, µ - Service rate of server, ꝭ - Utilization factor of the server,ʎeff -
Effective rate of arrival of customers ,M - Poisson distribution, N - Maximum numbers of
customers permitted in the system. Also, it denotes the size of the calling source of the
customers. GD - General discipline for service. This may be first in first – serve (FIFS), last-in-
first serve (LIFS) random order (Ro) etc.
1.7 Traffic intensity (or utilization factor)
An important measure of a simple queue is its traffic intensity given by Traffic intensity ꝭ =
Mean arrival time = ʎ (< 1) Mean service time µ and the unit of traffic intensity is Erlang
1.8 Classification of Queueing models
Generally, queueing models can be classified into six categories using Kendall’s notation with six
parameters to define a
model. The parameters of this notation are
P- Arrival rate distribution ie probability law for the arrival /inter – arrival time.
Q - Service rate distribution, ie probability law according to

12. which the customers are being served.
R - Number of Servers (ie number of service stations) X - Service discipline Y - Maximum number
of customers permitted in the system. Z - Size of the calling source of the customers.
A queuing model with the above parameters is written as (P/Q/R : X/Y/Z)
1. Model 1 : (M/M/1) : (GD/ ᴔ / ᴔ) Model
In this model (i) the arrival rate follows poisson (M) distribution. (ii) Service rate follows
poisson distribution (M) (iii) Number of servers is 1 (iv) Service discipline is general disciple (ie
GD) (v) Maximum number of customers permitted in the system is infinite (ᴔ) (vi) Size of the
calling source is infinite (ᴔ)
The steady state equations to obtain, Pn the probability of having customers in the system and
the values for Ls, Lq, Ws and Wq are given below.
n= 0,1,2,---- ᴔ.
Ls – Average number of customers waiting in the system (ie waiting in the system)
Ls = ʎ/ (µ- ʎ)
Lq – Average number of customers waiting in the queue (ie waiting in the queue)
Lq = ʎ^2/(µ(µ-ʎ)
Pn = ꝭ^n (1-po)
Average waiting time of customers in the system (in the queue and in the service station) =
Ws = Ls / ʎ
Wq = Average waiting time of customers in the queue. = Lq / ʎ .
PROBLEM 1:-The arrival rate of customers at a KFC restuarent follows a poisson distibution with
a mean of 30 per hours. The service rate of the waiter also follows poisson distribution with
mean of 45 per hour.
a) What is the probability of having zero customer in the system ?
b) What is the probability of having 8 customer in the system ?

13. c) What is the probability of having 12 customer in the system ?
d) Find Ls, Lq, Ws and Wq
Solution
Given arrival rate follows poisson distribution with mean ʎ= 30 customers per hour
Given service rate follows poisson distribution with mean µ= 45 customer Per hour
Utilization factor ꝭ = ʎ / µ = 30/45 = 2/3 = 0.67
a) The probability of having zero customer in the system P0 = (1- ꝭ ) = 1-0.67 =0.33
b) The probability of having 8 customer in the system
P8 = ꝭ^8 *P0 =(0.67)^8 *0.33=0.01340
c) The probability of having 12 customer in the system
P12= ꝭ^12 *P0 = (0.67)^12 *0.33=0.002859
d)
Ls – Average number of customers waiting in the system (ie waiting in the system)
Ls = ʎ/ (µ- ʎ) = 30/(45-30) =2 customers
Lq – Average number of customers waiting in the queue (ie waiting in the queue)
Lq = ʎ ^2 / µ (µ - ʎ) = 30^2/(45(45-30)) = 1.33
Average waiting time of customers in the system
Ws = Ls/ ʎ = 2/30 =1/15 = 0.0666
Average waiting time of customers in the queue
Wq = Lq/ ʎ = 1.33/30 =0.04433
PROBLEM 2. A restaurant has only are cashier during the peak hours customer arrive at the
rate of 20 customer per hour,the average number of customer that can be processed by cashier
is 30 per hour.
Calculate (1) probability that cashier is idle .
(2) average number of customer in the queuing system.
(3) average number of customer spends in the system.

14. (4) average number of customer in the queue.
(5) the average number of customer spends in queue waiting for service. ?
Solution:---
According to the information
Mean arrival rate, ʎ=20 cust/hr.
Mean service rate, µ=30 cust/hr.
(1) Probability that cashier is idle.
1-( ʎ/ µ) = 1-(20/30)=0.33
(2) average number of customer in the queuing system.
Ls = ʎ/ (µ- ʎ)= 20/(30-20) = 20/10 = 2 customer
(3) Average number of customer spends in the system.
Ws = 1/ (µ- ʎ)= Ls/ ʎ = 1/(30-20) =0.1 hour or 6 minutes
(4) average number of customer in the queue.
Lq = ʎ^2/(µ(µ-ʎ) = 20^2/(30(30-20)) =1.33 customer.
(5) the average number of customer spends in queue waiting for service.
Wq = Average waiting time of customers in the queue.
Lq / ʎ = 1.33/20=1/15 hour or 4 minutes
SIMULATED RESULTS:
PROBLEM1:

15. PROBLEM2:

17. CONCLUSION:This project invoked the use of waiting lines in the restuarent,as we know that on
Sunday or during parties there are too much rush in the restaurant so this problem can be
solved by using queuing theory.
REFERNCES:
https://www.youtube.com/watch?v=zYnyGUd9WSCkSD
http://www.prenhall.com/bp/app/russellc./queueing5.HTM
http://vixra.org/pdf/1008.0032v1.pdf
http://www.iasj.net/iasj?func=fulltext&aId=26659
http://www.slideshare.net/juliuscuaresma/queueing
http://www.ripublication.com/iraer-spl/iraerv4n5spl_03.pdf